The average SAT math score for high school seniors who graduated in 2005 was 520, which is the highest average ever and two points higher than the average for the class of 2004. According to Gaston Caperton, the president of the College Board, the high average is due to more students taking more advanced mathematics classes, such as calculus, in high school. Others say the increase may be due to more students taking SAT-preparation courses. Jon Reider, director of college counseling at San Francisco University High School, is quoted in The Christian Science Monitor as saying that the attention given to testing and test preparation is, "an enormous drain on kids' attention and focus, which should be on school."

Khipus (or quipus) are assemblages of strings and knots that are known to have represented numbers in the Incan civilization. Archaeologist Gary Urton and mathematician Carrie Brezine have written an article to show that khipus were also used as a language. Until now, the Incas are the only sufficiently complex civilization not to have a written language. Only about 700 khipus exist (most were destroyed by the conquistadors), so deciphering khipus has been difficult. Urton and Brezine believe that they have identified the first word in a khipu---the site where a particular khipu was found.

"Science and Society," hosted by David Lemberg and Samuel R. Kephart. World Talk Radio, 31 August 2005.

Guests on this program were Ryne Raffaelle, NanPower Research Laboratories, RIT; Douglas N. Arnold, Institute for Mathematics and its Applications (IMA), University of Minnesota; and Brian Rice, University of Dayton Research Institute. Arnold is director of the IMA, which has recently been awarded a 5-year, $19.5 million grant from the NSF--the largest grant the NSF has ever awarded for mathematics research. He responds to questions about the IMA's mission and activities, which aim to increase the impact of mathematics by fostering interdisciplinary research that links math with important problems from other disciplines. One example is imaging science, used in medical science and satellite photography. Arnold also notes that basic mathematics research often restructures the thinking process, which may result in applications much later. One example is materials science, in which research done in the early 1980s and 1990s continues to find applications. He describes the IMA's postdoctoral programs and summer workshops on modeling for graduate students (in which a representative from an industry presents a problem and teams of students conduct intensive research and present their results). The interview concludes with the host's invitation to interview Arnold again.

The basic mathematical model of statistical mechanics proposed by Ludwig Boltzmann and developed mathematically by Josiah Willard Gibbs has been a mainstay of physics and applies to many physical systems. But there are many important systems where the Boltzmann-Gibbs model cannot be used. For this reason, Mexican physicist Constantino Tsallis came up with a new variant on the classical model in the 1980s. Tsallis' model incorporates the expression pq in such a way that if q is set equal to 1, the original Boltzmann-Gibbs model is recovered. But if q is different from one, Tsallis's formula produces something new. "This led Tsallis to a new definition of entropy," Buchanan writes. "He called it q entropy." When he first came up with this new formula, Tsallis "had no idea what q might signify, but the way this new entropy worked mathematically suggested he might be on to something." Researchers have since shown that Tsallis's mathematics does indeed fit a variety of phenomena. However, the physical meaning of the parameter q has yet to be found. For this reason, many physicists remain skeptical that Tsallis's work represents any really new physical insights. But others are hopeful that this attempt to go beyond the Boltzmann-Gibbs model will eventually prove fruitful.

After some delay, Proof, the film version of David Auburn's Pultizer Prize-winning play, has been released. The reviews praise the cast (Gweneth Paltrow, Anthony Hopkins, Hope Davis, Jake Gyllenhaal) and the faithfulness to the screenplay, but few focus on the role of mathematics. The emphasis is on the relationships between the brilliant but disturbed mathematician father and his caretaker daughter Claire, between Claire and one of her father's students, and between Claire and her sister. The Reuters review covers the mathematics component more than the other reviews: "The question hovering over this quandary is whether mental illness, like talent, can be inherited. Claire, who gave up math studies to care for her beloved father, shows signs of both. The film also dips into several of the provocative myths about math. Is it really a young man's profession?... Are mathematicians predisposed to mental instability?..." The film was also reviewed after premieres in Montreal, New York, and Los Angeles. See: New Yorker ("Unhappy Families," by Anthony Lane, 19 September);MSNBC.com ("'Proof' is one of Paltrow's best," by John Hartl, 15 September);New York Times ("Nature or Nurture? You Do the Math," by Sylviane Gold, 11 September).

Consider the number of stars in our galaxy: 1011. Multiply this by 1 million, and there are more than this number of prokaryotes (or prokaryotic cells) ... in a ton of soil! But how diverse is this population? In the 25 August 2005 issue of Science, Thomas P. Curtis and William T. Sloan discuss the issue of microbial diversity in general and a paper published on page 1387 of this issue by Jason Gans, Murray Wolinsky, and John Dunbar in particular. Curtis and Sloan begin by explaining an existing technique---involving the "melting" and cooling of soil DNA---that has been used to estimate prokaryotic diversity in soil. At the same time, they describe how the work of conventional ecology to determine species abundance distributions has been applied to this microbiological concern. They also point out some results and limitations of these methods.

Curtis and Sloan then discuss Gans, Wolinsky, and Dunbar's recent work: How, by applying "new mathematical treatment to the [existing] data, the authors generate abundance curves, the most plausible of which suggests that there could be 107 distinct prokaryote taxa in 10 grams of pristine ... soil." (This is almost three orders of magnitude greater than previously thought.) "Moreover, rare organisms comprise most of this diversity." Gans, Wolinsky, and Dunbar also observed one important consequence of this distribution: Metal-contaminated soil samples displayed a more than 99.9 percent reduction in prokaryote diversity. Curtis and Sloan point out that this is not the end of the road, however, but just a roadmap for "the rational exploration of this frontier."

Simon Singh (author of Fermat's Last Theorem, The Code Book, and other popular science books) is launching a new radio program called "A Further Five Numbers," to air Tuesday mornings on BBC Radio 4 starting 23 August. He is "on a mission to shed a more positive light on maths." The article quotes Singh on some interesting facts about the numbers 1, 6, and 1,729. Hear the archived broadcasts.

Reporter Hirth notes the shortage of U.S. scientists, mathematicians and engineers to compete in today's world, and she finds a number of women to speak about their inspirations and experiences as scientists. (Hirth reports the U.S. Labor Department figures: Women make up only 10% of the nation's engineers, 29% of mathematicians and computer scientists, and 37% of life and physical scientists.) Mathematician Bettye Ann Case ("the first female full professor at [Florida State University]" and author of Complexities: Women in Mathematics) admits she "pounces" on strong applications from women interested in a graduate math program. Most of the women mentioned overcame---or continue to deal with---some obstacle or other, but benefited from having role models, mentors, and collaborative learning environments. Hirth hears an encouraging response to the question, "What advice do these scientists have for other women?": "Take as much math as possible."

This brief piece appears as a sidebar with a longer article, "Pies in the sky", about futuristic food. A French expert in "molecular gastronomy" has developed a symbolic notation, similar to mathematical notation, that can be used to encode recipes. Some symbols represent ingredients, others represent how the ingredients interact, superscripts indicate repeated actions, and so forth. The longer article speculates that one day machines might be able to "mathematically evolve several generations of new foods". But don't throw out the potato-masher just yet. As one person quoted in the article put it, "I'm not convinced that this will replace the need for chefs and handmade food."

The New York Times, CNN.com and local newspapers such as The Providence Journal published the results of a recent AP-AOL News Poll (of 1,000 adults, conducted August 9-11, 2005) which showed that people in the U.S. have a love-hate relationship with math. Many--37% of adults polled--recall they hated math in school ("twice as many people said they hated math as said that about any other subject," while about 23% said math was their favorite school subject). More than half said that children should get more education in both science and the arts and 15% said they wished they learned more about or studied more math in school (second after foreign language). The AP reports that "compared with students overseas, students in this country tend to be stronger in math in elementary school and move progressively behind as they get into high school," when they lag behind students of the same age in Europe and Asia. The AP interviewed a couple of teachers and individuals who think that students interested in math need to be nurtured, and that societal attitudes towards math need to change so that--as the report points out--the U.S. can cultivate a population literate in math--"critical in research, innovation and economic competitiveness." Ipsos News Center has published the Back to School Study.

"Fight for the naked quark," by Dana Mackenzie. New Scientist, 13 August 2005, pages 32-35.

"For 30 years, researchers have been battling to make sense of the strong force---the glue that sticks protons and neutrons together in nuclei," Mackenzie writes. "Yet their best mathematical theory has been so complex that no one could turn the dense equations into useful predictions." This hurdle now seems to be falling, as a result of more powerful computers and a handful of mathematical tricks that have rendered the calculations more tractable. A milestone was reached at the end of June 2005, when a particle accelator finished a series of experiments to estimate a quantity associated with a particle called a D-meson. Theorists had programmed a supercomputer to estimate that same quantity using a mathematical model and hours of number-crunching time. When the researchers found the answers matched, "the theorists were over the moon." The theory driving the calculations is called lattice quantum chromodynamics (QCD) and has for many years been "considered a poor relation of particle physics". After this year's triumph, theorists have high hopes for the power of lattice QCD.

"Searching for Intelligence in Edinburgh," by Daniel Winterstein. The Register, 10 August 2005.

The recent International Joint Conference in Artificial Intelligence (IJCAI) meeting drew 1,000 computer scientists, mathematicians, psychologists, and philosophers to Edinburgh. While there have been major advances in algorithms that underlie AI, the dream of machine intelligence remains just that, a dream. Geoff Hinton received an award for Research Excellence for his work on neural networks. Winterstein describes the history and applications of these programs, as well as statistical methods, Bayesian methods, and Markov models used in the field of AI. He concludes that despite the impressive appearance of some new Sony QRIO robots, "the only true intelligence remains our own."

Peter Donovan, a mathematician at the University of New South Wales, "says he is the first to identify the truth about Japan's operational code JM-25 [in WWII] since archives became available in 1975... that the cipher used groups of numbers that were multiples of three." This ultimately allowed the U.S., Australia, and Britain to break the code, not before Pearl Harbor in 1941, but in the Coral Sea Battle of 1942 and in the Battle of Midway in 1942. Skatssoon provides some background on code-breaking in the Pacific, and notes that Donovan's paper was published in Cryptologia.

Who is the secretive and elusive mathematician Shalosh B. Ekhad? And what does he have to do with the far more outgoing Rutgers mathematician Doron Zeilberger, whose web page is famous for its outrageous opinions? You'll have to read Szpiro's article to find out.

This article discusses the landmark result of Agrawal, Kayal, and Saxena in 2002, which provided a new, fundamentally faster method for testing whether or not a given number is prime. When the numbers are large, this is no trivial task. That prime numbers are the basis for codes that allow secure communications for everything from internet commerce to military secrets adds a "frisson of excitement to any discovery that relates primes to computation", Stewart writes. These codes are based on the difficulty of factoring numbers that are the product of large primes. The new primality test does not directly threaten the security of such codes, because it does not furnish a way to factor numbers. Nevertheless, the Agrawal-Kayal-Saxena result came out of left field and surprised many people who thought that primality testing was fundamentally more difficult. "Now they are not quite as sure as they used to be about factorization," Stewart notes.

Mathematician Alexandre J. Chorin (UC Berkeley) and colleagues have developed a new mathematical model that "suggests that droplets whipped from the tops of waves increase the speeds of winds well above what they'd be if the spray wasn't there." The articles explain the basics of the scientific experiements, the findings of which were published in the 9 August 2005 issue of Proceedings of the National Academy of Scinces. Science News includes a historical note, Chorin's speculation that perhaps some sailors, even in ancient times, cast oil on the sea during stormy weather to coat the water and reduce the formation of spray, thus calming the wind around the ship. Some physicists concede that spray may affect wind speed, but of course it's difficult to measure the spray and determine its effect during an actual hurricane.

Geeks. Nerds. Number crunchers who can't even tie their own shoes. The general public carries a variety of stereotypes about people who study theoretical mathematics, and while "cool" isn't among them, tales of the eccentricities of great mathematicians have endeared the profession to the masses. Many mathematicians think that outsiders view their field in a predominantly negative light, as evidenced by their comparatively small slice of the research funding pie and caused by both their removal from the applications of their theories and the stories of the brilliant but socially isolated masters of their discipline. The public, however, thirsts for such esoteric anecdotes about "supernatural intellectual powers," creating an image that is just as, if not more, beloved than that of other scientific areas.

It was not a typical mathematics meeting that took place 12-15 July 2005, on the Greek island of Mykonos. Encouraged by the appearance of mathematics and mathematicians in recent popular culture, Apostolos Doxiadis, author of Uncle Petros and the Goldbach Conjecture, invited a group of mathematicians, historians, artists, philosophers, playwrights, and novelists to come together to investigate the "common ground" between mathematics and storytelling. As Tomlin put it, participants wanted to know "What can professional mathematicians learn from the writers who are now taking an interest in their work?"

Speakers addressed such questions as whether "narrative approaches [can] help the increasingly esoteric sub-fields of mathematics communicate with one another," noted Tomlin. Some speakers gave examples of how telling a story about a problem, or finding an application of a problem, helped them to understand or solve those problems. The challenges of communicating mathematics to the public were also raised. In fact, Tomlin wrote that "some of the most heated discussions ... were about the mistakes in a recent popular book on infinity, and whether they really matter". University of California statistician Persi Diaconis commented, "To communicate we have to lie. If we don't we're deadly boring."

"Putting certainty in the bank," by Patrick Hayden. Nature, 4 August 2005.

In this article, researchers Michal Horodecki, Jonathan Oppenheim, and Andreas Winter present their work on prior quantum information: "given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system." Specifically, they discuss "conditional entropy," the partial information a system needs given its prior information, and provide an "operational meaning" for negative-valued conditional entropy, a counter-intuitive situation which cannot occur in the classical case, but can in the quantum world. In their own words, "if the partial information [a system needs] is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, then sender and receiver instead gain the corresponding potential for future quantum communication."

For those whose knowledge of quantum information theory is closer to a qubit, McGill University assistant professor of computer science Patrick Hayden provides an introduction to the topic and an overview of Horodecki, Oppenheim, and Winter's article in the same issue of Nature. Hayden concludes that, "in addition to finally placing the quantification of uncertainty in quantum mechanics on a solid footing, the new result opens the door to solving many previously intractable problems in quantum information theory."